#196580

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
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	C₂×C₄²	C₂⁴
prime radical	#2615	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

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, 103776);