#2309518

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=gh,d^2=gi,e^2=fg,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, 2216714);