#6457621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364817);