#354400

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306950, #72307310, #72307328
maximal normal subgroup of	#72306950, #72307310, #72307328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261596);