#368230

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2404
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1490 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)(7,12)(8,21,23,32)(9,15)(10,16)(11,17)(13,24,31,25)(14,20)(18,29,34,36)(19,28)(22,30,35,27)(26,33);(1,3,4,2)(5,7)(6,12)(8,22,23,35)(9,26)(10,19)(11,20)(13,29,31,36)(14,17)(15,33)(16,28)(18,25,34,24)(21,27,32,30);(5,8,10,24,11,23,9,25)(6,13,16,21,17,31,15,32)(7,18,19,35,20,34,26,22)(12,27,28,36,14,30,33,29);(5,7)(6,14)(8,18)(9,26)(10,19)(11,20)(12,17)(13,30)(15,28)(16,33)(21,29)(22,25)(23,34)(24,35)(27,31)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318131, #72318301, #72318313
maximal normal subgroup of	#72318131, #72318301, #72318313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275426);