#789591

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3084 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)(5,6,10,9)(7,11,8,12)(13,14,18,17)(15,20,16,19)(21,22,26,27)(23,24,25,28);(6,9)(11,12)(13,14,18,17)(15,19,16,20)(22,27)(23,25);(5,7,10,8)(6,11,9,12)(13,15,18,16)(14,19,17,20)(21,23,26,25)(22,28,27,24);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16,18,15)(14,20,17,19)(21,24)(22,23)(25,27)(26,28);(5,9,10,6)(7,12,8,11)(13,17,18,14)(15,20,16,19)(21,25,26,23)(22,28,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 696787);