#1195933

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2984 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)(6,7)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,25,26,23)(20,28,30,24)(27,31,29,32);(11,15)(13,16)(18,22)(19,26)(24,28)(27,29);(1,2,4,3)(5,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31);(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29);(1,3,4,2)(5,7,8,6)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,c^2=fgh,d^2=hi,e^2=fg,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, 1103129);