#904622

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2040 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=fgi,d^2=fg,e^2=fgi,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, 811818);