#194112

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 101308);