#650685

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

3030 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)(5,6,9,10)(7,8,11,12)(13,14)(15,16)(17,18)(19,20)(21,22,25,26)(23,24,27,28);(6,10)(8,12)(14,18)(16,19)(22,26)(23,27);(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,28)(26,27);(5,8)(6,11)(7,10)(9,12)(13,16,17,19)(14,15,18,20)(21,23)(22,24)(25,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 557881);