#196429

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103625);