#394555

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2232 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)(5,6,10,14)(7,11,16,26)(8,12,20,30)(9,21,24,13)(15,33,34,25)(17,31,22,27)(18,28,35,36)(19,32,23,29);(1,2)(3,4)(5,7)(6,11)(8,18)(9,22)(10,16)(12,28)(13,31)(14,26)(15,19)(17,24)(20,35)(21,27)(23,34)(25,29)(30,36)(32,33);(1,3,4,2)(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,32)(14,30)(16,34)(17,35)(18,22)(19,24)(21,29)(26,33)(27,36)(28,31);(5,7)(6,11)(8,15)(9,17)(10,16)(12,25)(13,27)(14,26)(18,19)(20,34)(21,31)(22,24)(23,35)(28,29)(30,33)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318058, #72318250, #72318301
maximal normal subgroup of	#72318058, #72318250, #72318301

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301751);