#3135118

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 2097152
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3042314);