#251852

order	512
nilpotency class	3
isoclinism	isoclinic class of #251827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

946 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341008, #72341025, #72341036, …
maximal normal subgroup of	#72341008, #72341025, #72341036, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159048);