#950628

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2206 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,20)(22,23)(25,26,30,29)(27,32,31,28);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,22,24,23)(25,26,30,29)(27,32,31,28);(1,2,4,5)(3,6,8,7)(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,22,20,23)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,31,29,27);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=i,e^2=fh,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, 857824);