#394202

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2570 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:

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

Polycyclic presentation:

2^9|b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,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, 301398);