#5957645

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3696 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)(3,7)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(25,26,28,31)(27,30,32,29);(2,5)(6,8)(17,19,21,24)(18,20,23,22)(25,27,28,32)(26,29,31,30);(9,11,12,16)(10,13,14,15)(17,19,21,24)(18,22,23,20)(25,26,28,31)(27,30,32,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326954, #72326982, #72327022
maximal normal subgroup of	#72326954, #72326982, #72327022
maximal quotient of	#72328844

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5864841);