#1080495

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 987691);