#395505

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×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2213 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)(7,11)(8,18,19,12)(9,13)(10,14)(15,23)(16,24,30,31)(17,25)(20,32,33,26)(21,27)(22,28)(29,35,34,36);(1,2,3,4)(7,15)(9,21)(11,23)(13,27)(16,29)(17,22)(20,33)(24,35)(25,28)(26,32)(30,34)(31,36);(5,7)(6,11)(8,16)(9,22)(10,17)(12,24)(13,28)(14,25)(15,21)(18,31)(19,30)(20,34)(23,27)(26,36)(29,33)(32,35);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,18)(15,29)(17,30)(21,33)(22,34)(23,35)(25,31)(27,32)(28,36)

Polycyclic presentation:

2^9|a^2=f,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, 302701);