#358298

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265494);