#2813853

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1782 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)(7,12)(8,9)(10,11)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(24,32)(35,36);(6,11)(9,12)(15,19)(18,20)(21,22,25,26)(23,30,29,27)(24,33,32,28)(31,35,34,36);(5,7,10,8)(6,12,11,9)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,33)(34,36);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(21,23,25,29)(22,27,26,30)(24,34,32,31)(28,36,33,35);(5,9,10,12)(6,8,11,7)(13,16,17,14)(15,20,19,18)(21,24,25,32)(22,28,26,33)(23,31,29,34)(27,35,30,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 2721049);