#393034

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
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

1798 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)(5,6,11,16,12,15,10,7)(8,19,22,29,23,17,21,13)(9,24,26,30,25,14,27,18)(20,31,34,36,33,32,35,28);(1,2,3,4)(5,7,11,15,12,16,10,6)(8,17,22,29,23,19,21,13)(9,18,27,24,25,30,26,14)(20,31,35,36,33,32,34,28);(5,8,10,21,12,23,11,22)(6,13,16,19,15,29,7,17)(9,20,26,34,25,33,27,35)(14,28,30,32,24,36,18,31);(5,9)(6,14)(7,18)(8,20)(10,26)(11,27)(12,25)(13,28)(15,24)(16,30)(17,31)(19,32)(21,34)(22,35)(23,33)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^b=gi,h^b=hi,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, 300230);