#153947

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=eh,b^a=be,c^a=cf,c^b=cg,d^c=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61143);