#6380104

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2473 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287300);