#6403693

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310889);