#6393492

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×#3541	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3228 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=i,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, 6300688);