#220070

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
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

1056 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)(5,6,9,13,10,14,7,11)(8,16,15,20,18,12,17,19)(21,22)(23,30)(24,28)(25,33)(26,29)(27,32)(31,36)(34,35);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,17,18,15)(12,20,16,19)(23,31)(27,35)(30,36)(32,34);(5,8,10,18)(6,12,14,16)(7,15,9,17)(11,19,13,20)(21,23,26,31)(22,27,29,35)(24,32,25,34)(28,36,33,30);(2,4)(5,7,10,9)(6,13,14,11)(8,15,18,17)(12,20,16,19)(21,24,26,25)(22,28,29,33)(23,32,31,34)(27,36,35,30)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127266);