#2813856

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

2100 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)(7,12,8,11)(13,14)(15,19)(16,18)(17,20)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,35,34,36);(5,7,10,8)(6,11,9,12)(13,14,16,18)(15,17,20,19)(22,26)(23,30)(24,33)(35,36);(5,7,10,8)(6,12,9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,33)(34,36);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15,16,20)(14,17,18,19)(21,23,25,30)(22,27,26,29)(24,34,33,31)(28,36,32,35);(5,9,10,6)(7,11,8,12)(13,14)(15,19)(16,18)(17,20)(21,24,25,33)(22,28,26,32)(23,31,30,34)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fh,d^2=f,e^2=fgh,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, 2721052);