#2035387

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

2178 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)(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22,26,29)(23,27,24,30)(25,33,34,28)(31,36,32,35);(6,10)(8,12)(14,18)(16,19)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(13,14)(15,16)(17,18)(19,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8)(6,11)(7,10)(9,12)(13,16)(14,15)(17,19)(18,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=fh,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, 1942583);