#204607

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1138 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)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20)(21,22)(23,26,29,34)(24,31)(25,27)(28,32)(30,36,35,33);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19)(21,23)(22,26,27,34)(24,30)(25,29)(28,33,31,36)(32,35);(1,2,3,4)(6,11)(7,14)(12,16)(15,20)(22,27)(23,29)(28,31)(30,35);(21,24,25,32)(22,28,27,31)(23,30,29,35)(26,33,34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=i,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, 111803);