#5025702

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,10)(8,12)(9,11)(15,17)(16,19)(21,22,26,27)(23,28,30,24)(25,33,36,29)(31,32,34,35);(5,7,9,8)(6,10,11,12)(13,14)(15,16)(17,19)(18,20)(22,27)(23,30)(29,33)(31,34);(6,11)(10,12)(13,15,18,17)(14,19,20,16)(21,23)(22,28)(24,27)(25,34)(26,30)(29,32)(31,36)(33,35);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,18,19)(14,15,20,17)(21,24,26,28)(22,23,27,30)(25,35,36,32)(29,34,33,31);(5,8,9,7)(6,12,11,10)(13,17,18,15)(14,19,20,16)(21,25)(22,29)(23,31)(24,32)(26,36)(27,33)(28,35)(30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=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, 4932898);