#368222

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1427 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275418);