#1145764

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2836 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)(10,14)(12,15)(17,18,22,25)(19,26,23,20)(21,29,31,24)(27,28,30,32);(6,8)(17,19,22,23)(18,20,25,26)(21,30,31,27)(24,32,29,28);(1,2,4,3)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(1,2,4,3)(9,11,13,16)(10,15,14,12)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,29)(31,32);(1,3,4,2)(9,12,13,15)(10,16,14,11)(17,21)(18,24)(19,27)(20,28)(22,31)(23,30)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=fg,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, 1052960);