#2542308

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3342 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)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,23,24,21)(25,26,29,32)(27,30,28,31);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18,20,22)(19,21,24,23);(3,7)(6,8)(10,12)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,28)(26,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fhi,e^2=hi,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	#72300714, #72300747, #72300783, …
maximal normal subgroup of	#72300714, #72300747, #72300783, …
maximal quotient of	#72328554, #72328655, #72330486

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2449504);