#674121

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fgi,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, 581317);