#361383

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2280 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268579);