#368226

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 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)(7,11)(8,18,19,33)(9,13)(10,14)(12,20,26,34)(15,28,29,32)(16,23)(17,24)(21,27)(22,30,25,36)(31,35);(1,3,4,2)(5,7,10,17)(6,11,14,24)(8,15)(9,16,21,31)(12,25)(13,23,27,35)(18,32)(19,29)(20,30)(22,26)(28,33)(34,36);(5,8)(6,12,14,26)(7,15,17,29)(9,20)(10,19)(11,22)(13,18,27,33)(16,30,31,36)(21,34)(23,28)(24,25)(32,35);(9,21)(13,27)(16,31)(18,33)(20,34)(23,35)(28,32)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,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, 275422);