#1258553

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3283 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267303
maximal normal subgroup of	#72267303
maximal quotient of	#72267320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1165749);