#3078371

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2069 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)(6,9)(10,11)(14,18)(16,19)(21,22,26,24)(23,29,30,27)(25,28,33,32)(31,35,36,34);(5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,20,16)(22,24)(23,30)(25,33)(34,35);(5,7,8,12)(6,10,9,11)(13,15,17,20)(14,19,18,16)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(1,2)(3,4)(5,7)(6,11)(8,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,27,30,29)(25,28,33,32)(31,35,36,34);(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,20,18,15)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=g,e^2=fgh,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, 2985567);