#777880

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2264 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)(6,10)(7,12)(13,14,18,20)(15,16,17,19)(21,22,25,24)(23,27,28,26);(5,6,9,10)(7,11,12,8)(13,15,18,17)(14,16,20,19)(21,22,25,24)(23,27,28,26);(13,16,18,19)(14,15,20,17)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,16)(17,19)(18,20)(21,22,25,24)(23,26,28,27);(5,8,9,11)(6,12,10,7)(13,17,18,15)(14,19,20,16)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=h,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, 685076);