#393031

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1894 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)(7,15)(8,16)(9,22,25,13)(10,18)(11,14)(12,20)(17,23,28,24)(19,32,35,30)(21,31)(26,34,36,33)(27,29);(1,2,3,4)(5,7,15,6,11,18,10,14)(8,16,29,12,21,31,20,27)(9,17,23,13,25,28,24,22)(19,30,33,26,35,32,34,36);(5,8,15,29,11,21,10,20)(6,12,18,31,14,27,7,16)(9,19,23,33,25,35,24,34)(13,26,28,32,22,36,17,30);(5,9)(6,13)(7,17)(8,19)(10,24)(11,25)(12,26)(14,22)(15,23)(16,30)(18,28)(20,34)(21,35)(27,36)(29,33)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,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.

Relationships

maximal subgroup of	#72281002, #72281182, #72281320, …
maximal normal subgroup of	#72281002, #72281182, #72281320, …
maximal quotient of	#72313502, #72343138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300227);