#528596

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=ghi,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 435792);