#6817034

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	#25762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fhi,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, 6724230);