#2960815

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

2414 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)(3,6)(4,7)(5,8)(10,11)(12,16)(17,18)(19,23)(20,22)(21,24)(27,28)(30,31);(1,3,4,8)(2,5,7,6)(10,11)(14,15)(17,19,21,20)(18,22,24,23)(25,26,29,32)(27,30,28,31);(9,10,13,11)(12,14,16,15)(17,19,21,20)(18,23,24,22)(25,27,29,28)(26,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(17,20,21,19)(18,23,24,22)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2868011);