#196976

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1302 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)(5,6,9,14,10,15,8,13)(7,12,16,19,18,20,17,11)(21,22)(23,27,32,35)(24,28)(25,29)(26,30)(31,34,33,36);(5,7)(6,11,15,19)(8,17)(9,16)(10,18)(12,13,20,14)(21,23)(22,27,30,35)(24,31)(25,33)(26,32)(28,34,29,36);(2,4)(5,7)(6,12)(8,17)(9,16)(10,18)(11,13)(14,19)(15,20)(21,24,26,25)(22,28,30,29)(23,31,32,33)(27,34,35,36);(5,8,10,9)(6,13,15,14)(7,16,18,17)(11,12,19,20)(21,25,26,24)(22,29,30,28)(23,31,32,33)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,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, 104172);