#613094

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3108 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)(7,11)(10,12)(14,17)(16,19)(22,25)(26,28);(5,6,9,8)(7,10,11,12)(13,14,18,17)(15,19,20,16)(22,25)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15,18,20)(14,19,17,16)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,20)(18,19);(5,8,9,6)(7,12,11,10)(13,17,18,14)(15,16,20,19)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 520290);