#521155

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df

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, 428351);