#6372974

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,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, 6280170);