#2228035

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2001 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fi,d^2=f,e^2=fg,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, 2135231);