#368734

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1763 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316311, #72316326, #72316345
maximal normal subgroup of	#72316311, #72316326, #72316345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275930);