#2305995

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1904 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fghi,c^2=fghi,d^2=fgh,e^2=fgh,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, 2213191);