#6370854

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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

4510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,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, 6278050);