#2892814

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

2264 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=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, 2800010);