#2623254

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
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	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2928 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232579, #72232633, #72232687
maximal normal subgroup of	#72232579, #72232633, #72232687

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2530450);