#725959

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3352 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330845, #72330855, #72330927
maximal normal subgroup of	#72330845, #72330855, #72330927
maximal quotient of	#72333141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 633155);