#196584

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1192 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)(5,6,9,11,10,14,8,13)(7,12,15,18,16,20,17,19)(22,27,29,33)(28,34,36,35);(6,11,14,13)(12,18,20,19)(21,22,25,33,26,29,24,27)(23,28,30,35,31,36,32,34);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,23)(22,28)(24,32)(25,30)(26,31)(27,34)(29,36)(33,35);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(21,24,26,25)(22,27,29,33)(23,30,31,32)(28,35,36,34)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,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, 103780);