#836689

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,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.

Relationships

maximal subgroup of	#72336442, #72336449, #72336487, …
maximal normal subgroup of	#72336442, #72336449, #72336487, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 743885);