#394278

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301474);