#528698

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

1910 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)(7,9)(8,11)(10,12)(13,14,18,20)(15,16,17,19)(21,22,25,24)(23,26,27,28);(6,7)(10,11)(13,15,18,17)(14,16,20,19)(23,27)(26,28);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,16,18,19)(14,15,20,17)(21,23,25,27)(22,26,24,28);(5,7,9,6)(8,11,12,10)(13,14)(15,16)(17,19)(18,20)(21,22,25,24)(23,26,27,28);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,19,20,16)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=h,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, 435894);