#840672

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fg,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 747868);