#687582

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 594778);