#7503226

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2785 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)(13,14,18,23,19,24,15,20)(16,27,25,35,30,38,29,21)(17,31,26,37,32,36,33,22)(28,34,41,43,42,44,39,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,24,23)(16,25,30,29)(17,26,32,33)(21,27,35,38)(22,31,37,36)(28,39,42,41)(34,40,44,43);(2,4)(13,16,19,30)(14,21,24,35)(15,25,18,29)(17,28,32,42)(20,27,23,38)(22,34,37,44)(26,39,33,41)(31,40,36,43);(6,10)(7,12)(13,17)(14,22)(15,26)(16,28)(18,33)(19,32)(20,31)(21,34)(23,36)(24,37)(25,39)(27,40)(29,41)(30,42)(35,44)(38,43);(13,15,19,18)(14,23,24,20)(16,29,30,25)(17,26,32,33)(21,27,35,38)(22,36,37,31)(28,41,42,39)(34,40,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^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, 7410422);