#1409068

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(10,12)(11,15)(17,18,22,25)(19,24,28,23)(20,29,31,21)(26,27,30,32);(6,8)(17,19,22,28)(18,23,25,24)(20,26,31,30)(21,27,29,32);(5,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,25)(26,31)(29,32);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,20,22,31)(18,21,25,29)(19,26,28,30)(23,27,24,32);(9,12,13,10)(11,16,15,14)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1316264);