#382134

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224752, #72224755, #72224758
maximal normal subgroup of	#72224752, #72224755, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289330);