#606823

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,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, 514019);