#6465839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#28225	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1951 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)(15,23)(16,21)(17,19)(18,30)(20,27)(22,25)(24,36)(26,35)(28,32)(29,41)(31,38)(33,40)(34,39)(37,43)(42,44);(2,3)(6,9)(10,12)(14,19)(15,24)(16,28)(22,30)(27,38)(29,39)(33,42)(35,43);(1,2,4,3)(13,15,25,38,17,24,18,27)(14,20,30,36,19,31,22,23)(16,26,39,44,28,37,29,40)(21,33,41,43,32,42,34,35);(1,3,4,2)(13,16)(14,21)(15,26)(17,28)(18,29)(19,32)(20,33)(22,34)(23,35)(24,37)(25,39)(27,40)(30,41)(31,42)(36,43)(38,44);(5,7,8,11)(6,10,9,12)(15,24)(16,28)(20,31)(21,32)(23,36)(27,38)(29,39)(34,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373035);