#6467429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6374625);