#687362

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2424	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3458 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22,25,26)(23,27,28,24);(6,9)(10,11)(14,18)(16,20)(21,23,25,28)(22,24,26,27);(7,12)(10,11)(14,18)(15,19)(22,26)(23,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^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, 594558);