#160353

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=egh,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 67549);