#195980

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1421 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)(5,6)(7,10,14,17)(8,11)(9,12)(13,16,19,20)(15,18)(21,22,24,28,25,27,32,33)(23,26,30,35,29,34,31,36);(5,7)(6,10,12,17)(8,13)(9,14)(11,16,18,20)(15,19)(21,23,24,31,25,29,32,30)(22,26,33,35,27,34,28,36);(2,4)(23,29)(26,34)(30,31)(35,36);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,16,17,20)(22,27)(26,34)(28,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,e^b=eh,g^b=gh

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, 103176);