#606757

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

2034 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=g,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, 513953);