#606802

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

2170 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)(14,17)(18,20)(22,26)(23,29)(24,31)(34,36);(6,7)(10,11)(15,19)(18,20)(21,22,25,26)(23,27,29,33)(24,32,31,28)(30,36,35,34);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20);(5,7,9,6)(8,11,12,10)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,35)(32,36);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,31)(22,28,26,32)(23,30,29,35)(27,34,33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=f,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, 513998);