#360124

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,g^c=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, 267320);