#353553

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#381
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=be,c^a=cf,e^a=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260749);