#194901

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1268 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)(5,6,9,13,10,14,8,12)(7,15,18,11,16,20,17,19)(21,22,26,30,27,32,25,31)(23,33,35,28,29,36,34,24);(1,2,3,4)(5,7)(6,11)(8,18)(9,17)(10,16)(12,20)(13,15)(14,19)(21,23,27,29)(22,28,32,24)(25,35,26,34)(30,33,31,36);(21,24,27,28)(22,29,32,23)(25,33,26,36)(30,35,31,34);(7,16)(11,19)(15,20)(17,18)(21,25,27,26)(22,30,32,31)(23,34,29,35)(24,33,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102097);