#386092

order	512
nilpotency class	3
isoclinism	isoclinic class of #219696
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

1679 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)(7,13,19,28)(8,20,23,12)(9,24)(10,15)(11,17)(14,26)(16,21)(18,32,33,29)(22,27,36,34)(25,31)(30,35);(1,2)(3,4)(5,7)(6,12)(8,21)(9,22)(10,23)(11,19)(13,16)(14,29)(15,28)(17,20)(18,25)(24,32)(26,36)(27,30)(31,34)(33,35);(5,8,21,7,11,23,10,19)(6,13,15,20,17,28,16,12)(9,22,35,18,26,36,25,33)(14,29,30,34,24,32,31,27);(1,3,4,2)(5,9,21,35,11,26,10,25)(6,14,16,31,17,24,15,30)(7,18,23,36,19,33,8,22)(12,27,28,32,20,34,13,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250486, #72256516, #72256521
maximal normal subgroup of	#72250486, #72256516, #72256521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293288);