#369188

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226670, #72226675, #72226677
maximal normal subgroup of	#72226670, #72226675, #72226677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276384);