#2076724

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3792 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1983920);