#862628

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1868 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)(4,8)(5,6)(9,10,13,16)(11,12,15,14)(25,26,30,28)(27,29,32,31);(2,6)(4,7)(9,10,13,16)(11,12,15,14)(18,21)(22,24)(25,27,30,32)(26,31,28,29);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,14,16,12)(17,18,20,21)(19,22,23,24)(25,28,30,26)(27,29,32,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(25,29)(26,27)(28,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,13,14)(10,15,16,11)(17,19,20,23)(18,22,21,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 769824);