#160352

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

2668 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)(13,14,17,15)(16,19,20,18)(22,25)(23,27);(2,4)(6,7)(10,11)(16,20)(18,19)(21,22,24,25)(23,28,27,26);(6,7)(9,10,12,11)(13,15,17,14)(16,19,20,18)(21,23,24,27)(22,26,25,28);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(13,16)(14,18)(15,19)(17,20)(21,22)(23,28)(24,25)(26,27)

Polycyclic presentation:

2^9|a^2=fh,b^2=efg,c^2=fgh,d^2=eg,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, 67548);