#2555280

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3009 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300839, #72300852, #72300896
maximal normal subgroup of	#72300839, #72300852, #72300896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2462476);