#192654

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99850);