#6817303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2037 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6724499);