#522203

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

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

2490 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)(5,6,9,7)(8,11,12,10)(13,14,17,18)(15,20,16,19)(23,24)(26,27);(1,3,4,2)(6,7)(10,11)(14,18)(19,20)(21,22,25,28)(23,26,24,27);(5,6,9,7)(8,10,12,11)(13,15,17,16)(14,19,18,20)(21,23,25,24)(22,26,28,27);(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24,25,23)(22,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,15)(14,20,18,19)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=fg,e^2=f,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, 429399);