#1443613

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1350809);