#3069108

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3198 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)(11,14)(12,16)(13,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(6,8)(9,11,13,16)(10,14,15,12)(18,22)(20,28)(23,25)(26,32);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,25)(20,29)(21,22)(23,27)(24,28)(26,30)(31,32);(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^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, 2976304);