#2594064

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

2054 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2501260);