#6399555

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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×#546	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,21,23,28)(18,25,26,20)(19,27,22,24);(6,9)(7,11)(14,16)(19,22)(20,25)(21,28);(13,15,17,23)(14,19,16,22)(18,27,26,24)(20,28,25,21);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,22,23,19)(18,25,26,20)(21,24,28,27);(1,3,4,2)(6,9)(10,12)(14,16)(19,22)(20,25)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306751);