#369110

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
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

1906 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)(8,19,21,12)(9,13)(10,24)(11,17)(14,23)(16,29,33,25)(18,26)(20,27,36,34)(22,28,31,35)(30,32);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,20)(10,18,23,32)(12,27)(14,26,24,30)(16,31)(19,34)(21,36)(22,33)(25,35)(28,29);(5,8,10,22,9,21,23,31)(6,12,24,35,13,19,14,28)(7,16,32,36,17,33,18,20)(11,25,26,27,15,29,30,34);(1,3,4,2)(6,13)(11,15)(12,19)(14,24)(25,29)(26,30)(27,34)(28,35)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 276306);