#1104393

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2719 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:

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

Polycyclic presentation:

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

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, 1011589);