#194273

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1206 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)(5,6,9,13,10,14,8,11)(7,12,15,19,16,20,17,18)(21,22,25,30,26,28,24,29)(23,31,34,27,32,35,33,36);(1,2,3,4)(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,35)(30,31);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,35)(30,31);(5,8,10,9)(6,11,14,13)(7,15,16,17)(12,19,20,18)(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=i,g^2=h,b^a=be,c^a=cf,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101469);