#764596

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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

3778 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 671792);