#400995

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1784 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 308191);