#515284

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2310 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=h,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^c=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, 422480);