#6810205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717401);