#6372976

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280172);