#2036751

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

1972 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)(15,20)(16,19)(17,18)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36);(5,6,10,8)(7,12,11,9)(13,15,18,16)(14,19,17,20)(22,24)(25,33)(27,29)(30,36);(5,7)(6,9)(8,12)(10,11)(13,16,18,15)(14,19,17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,9,10,12)(6,11,8,7)(13,17,18,14)(15,19,16,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|b^2=fg,c^2=fh,d^2=gh,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, 1943947);