#395502

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
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

2109 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)(5,6,11,26,10,16,9,15)(7,12,19,29,20,14,18,28)(8,21,25,13,24,33,23,32)(17,34,22,27,36,30,35,31);(1,2,3,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,19)(13,30)(14,16)(15,28)(17,23)(21,27)(24,35)(25,36)(26,29)(31,33)(32,34);(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,27)(14,31)(15,33)(16,32)(18,22)(19,35)(20,36)(21,26)(28,30)(29,34);(5,7)(6,14)(8,17)(9,19)(10,20)(11,18)(12,16)(13,31)(15,28)(21,34)(22,23)(24,36)(25,35)(26,29)(27,32)(30,33)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 302698);