#371007

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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)(5,6,10,15)(7,16,20,25)(8,12,21,29)(9,13,17,14)(11,23,26,19)(18,33,34,31)(22,30,32,28)(24,35,27,36);(2,4)(5,7,17,23,10,20,9,19)(6,11,13,25,15,26,14,16)(8,18,32,36,21,34,22,35)(12,27,30,31,29,24,28,33);(5,8)(6,12)(7,18)(9,22)(10,21)(11,24)(13,30)(14,28)(15,29)(16,31)(17,32)(19,35)(20,34)(23,36)(25,33)(26,27);(2,4)(6,13,15,14)(8,21)(11,25,26,16)(12,28,29,30)(18,34)(22,32)(24,31,27,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255984, #72256186, #72256189, …
maximal normal subgroup of	#72255984, #72256186, #72256189, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278203);