#674132

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1969 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=gi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 581328);