#846566

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2934 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)(6,7)(8,12)(13,14,17,20)(15,16,19,18)(21,22)(23,27)(24,28)(25,26);(5,6,9,7)(8,10,12,11)(15,19)(16,18)(21,23,26,24)(22,27,25,28);(13,15,17,19)(14,18,20,16)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,17,18)(14,19,20,15)(21,25,26,22)(23,27,24,28);(5,8)(6,10)(7,11)(9,12)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 753762);