#195333

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1200 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)(5,6,10,14,11,7,9,13)(8,16,17,15,19,20,18,12)(21,22,25,30,26,28,24,29)(23,31,34,27,32,35,33,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,19)(16,17)(18,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,35)(30,31);(5,8)(6,12)(7,15)(9,18)(10,17)(11,19)(13,20)(14,16)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,35)(30,31);(5,9,11,10)(6,13,7,14)(8,17,19,18)(12,16,15,20)(22,28)(23,32)(29,30)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102529);