#2814031

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

2136 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)(5,6,10,9)(7,11,8,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,34)(35,36);(6,9)(11,12)(13,14,16,18)(15,17,20,19)(22,24)(25,33)(27,29)(30,36);(5,7,10,8)(6,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,15,16,20)(14,17,18,19)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(5,9,10,6)(7,12,8,11)(13,14)(15,19)(16,18)(17,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=f,e^2=fh,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, 2721227);