#369047

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1904 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)(7,15)(8,19,21,34)(9,13)(10,14)(11,17)(12,22,27,35)(16,30,31,33)(18,29)(20,28)(23,25)(24,36,26,32);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,16)(10,23,20,18)(12,26)(14,29,28,25)(19,33)(21,31)(22,36)(24,27)(30,34)(32,35);(5,8,20,35,9,21,10,22)(6,12,14,19,13,27,28,34)(7,16,18,32,17,31,23,36)(11,24,29,30,15,26,25,33);(1,3,4,2)(6,13)(11,15)(12,27)(14,28)(19,34)(24,26)(25,29)(30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276243);