#346292

order	512
nilpotency class	3
isoclinism	isoclinic class of #346284
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,5)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,33)(28,38)(29,40)(32,45)(34,37)(35,41)(36,43)(39,46)(42,48)(44,47);(1,3)(2,7,5,9)(4,10)(6,11)(8,15,12,14)(13,16)(17,19)(18,23,26,27)(20,28)(21,34)(22,30)(24,38,31,36)(25,40,41,37)(29,35)(32,42)(33,43)(39,48,45,47)(44,46);(1,4,6,13)(2,8)(3,10,11,16)(5,12)(7,14)(9,15)(17,20,22,33)(18,24)(19,28,30,43)(21,32,35,46)(23,36)(25,39)(26,31)(27,38)(29,44,34,42)(37,48)(40,47)(41,45);(1,5,6,2)(3,9,11,7)(4,8,13,12)(10,15,16,14)(17,21,22,35)(18,25,26,41)(19,29,30,34)(20,32,33,46)(23,37,27,40)(24,39,31,45)(28,42,43,44)(36,47,38,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=defh,e^b=eh,f^b=fhi,d^c=de,e^c=ei,f^c=fh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235375, #72235376, #72235378, …
maximal normal subgroup of	#72235375, #72235376, #72235378, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253488);