#607003

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1936 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514199);