#945626

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2431	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3902 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)(6,7)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27);(6,7)(10,12)(14,15)(19,21)(20,22)(26,27);(5,6,8,7)(9,10)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(1,2,3,4)(5,7,8,6)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 852822);