#612327

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2234 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519523);