#6396551

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 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)(6,9)(7,11)(13,14,17,19)(15,23,24,37)(16,21,28,32)(18,30,25,22)(20,33,31,27)(26,36,38,44)(29,41,39,35)(34,43,42,40);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(16,28)(22,30)(27,33)(29,39)(34,42)(36,44);(1,3,4,2)(13,15,25,33,17,24,18,27)(14,20,30,37,19,31,22,23)(16,26,39,43,28,38,29,40)(21,34,41,44,32,42,35,36);(6,9)(10,12)(13,16)(14,21)(15,26)(17,28)(18,29)(19,32)(20,34)(22,35)(23,36)(24,38)(25,39)(27,40)(30,41)(31,42)(33,43)(37,44);(5,7,8,11)(6,10,9,12)(14,19)(15,24)(16,28)(22,30)(27,33)(29,39)(34,42)(36,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,e^d=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, 6303747);