#2037631

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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

1995 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)(6,8)(9,12)(13,14,17,18)(15,19,20,16)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,6,10,8)(7,12,11,9)(14,18)(16,19)(22,26)(24,32)(27,29)(30,36);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,19,18,16)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,9,10,12)(6,11,8,7)(13,16,17,19)(14,20,18,15)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1944827);