#195232

order	512
nilpotency class	3
isoclinism	isoclinic class of #18316
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1438 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)(5,6,9,13,11,14,10,19)(7,15,16,20,8,17,18,12)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34);(1,2,3,4)(5,7,11,8)(6,12,14,20)(9,18,10,16)(13,17,19,15)(21,23)(22,26)(24,29)(25,30)(27,33)(28,34)(31,36)(32,35);(5,8,11,7)(6,12,14,20)(9,18,10,16)(13,15,19,17)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36);(22,28)(26,34)(27,31)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102428);