#6806669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6713865);