#395506

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 302702);