#220052

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1068 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)(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18)(21,22)(23,27)(24,32)(25,29)(26,30)(28,31)(33,36)(34,35);(1,2)(3,4)(7,14)(10,13)(15,16)(18,20)(21,23,26,25)(22,27,30,29)(24,33,34,31)(28,32,36,35);(5,7,9,14)(6,10,11,13)(8,16,17,15)(12,20,19,18)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,32)(30,36);(2,4)(6,11)(10,13)(12,19)(18,20)(21,23,26,25)(22,29,30,27)(24,31,34,33)(28,32,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127248);