#7503222

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410418);