#393029

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300225);