#6438349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2780 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,26,29,20)(16,22,32,39)(17,25,19,23)(21,33,38,31)(27,41,43,35)(28,36,30,37)(34,42,40,44);(1,2)(3,4)(6,9)(10,12)(13,15,18,29)(14,20,24,26)(16,27,32,43)(17,28,19,30)(21,34,38,40)(22,35,39,41)(23,36,25,37)(31,42,33,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,27)(17,31)(18,32)(19,33)(20,34)(22,23)(24,38)(25,39)(26,40)(28,42)(29,43)(30,44)(35,36)(37,41);(2,4)(13,16)(14,22)(15,27)(17,33)(18,32)(19,31)(20,35)(21,23)(24,39)(25,38)(26,41)(28,44)(29,43)(30,42)(34,36)(37,40);(13,17,18,19)(14,23,24,25)(15,28,29,30)(16,31,32,33)(20,36,26,37)(21,22,38,39)(27,42,43,44)(34,35,40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345545);