#346278

order	512
nilpotency class	3
isoclinism	isoclinic class of #346276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

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

normal subgroups only

This group has 607 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,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	#72235170, #72235171, #72235173, …
maximal normal subgroup of	#72235170, #72235171, #72235173, …
maximal quotient of	#72268764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253474);