#6816232

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723428);