#6440793

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2695 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)(13,14,19,24)(15,25,28,21)(16,20,30,35)(17,22,31,37)(18,33,26,23)(27,39,41,36)(29,34,42,44)(32,43,40,38);(1,2)(3,4)(6,9)(10,12)(13,15,26,30,19,28,18,16)(14,20,33,21,24,35,23,25)(17,27,40,42,31,41,32,29)(22,34,43,36,37,44,38,39);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,28,19,30,26,15)(14,21,23,20,24,25,33,35)(17,29,32,41,31,42,40,27)(22,36,38,34,37,39,43,44);(2,4)(13,17)(14,22)(15,27)(16,29)(18,32)(19,31)(20,34)(21,36)(23,38)(24,37)(25,39)(26,40)(28,41)(30,42)(33,43)(35,44);(17,31)(22,37)(27,41)(29,42)(32,40)(34,44)(36,39)(38,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6347989);