#204605

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1180 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:

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

Polycyclic presentation:

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

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, 111801);