#6396545

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×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(6,9)(7,11)(13,14,19,22)(15,17,21,18)(16,20,25,28)(23,26,27,24);(1,2,4,3)(5,6)(7,10)(8,9)(11,12);(1,3,4,2)(13,15,18,14,19,21,17,22)(16,23,24,20,25,27,26,28);(6,9)(10,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,25)(21,27)(22,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,15)(16,24,25,26)(20,23,28,27)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,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, 6303741);