#607004

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

1920 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)(7,9)(8,10)(11,12)(15,19)(18,20)(21,22,26,29)(23,27,31,24)(25,33,35,28)(30,32,34,36);(6,8)(9,11)(14,17)(18,20)(21,23,26,31)(22,24,29,27)(25,34,35,30)(28,36,33,32);(1,3,4,2)(5,7,10,12)(6,11,8,9)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,36)(32,33)(34,35);(5,8,10,6)(7,9,12,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,27)(22,23,29,31)(25,32,35,36)(28,30,33,34);(5,9)(6,7)(8,12)(10,11)(21,25,26,35)(22,28,29,33)(23,30,31,34)(24,32,27,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,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, 514200);