#299178

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fhi,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206374);