#773749

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=g,e^2=fg,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, 680945);