#299144

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	C₂×#3207	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1125 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,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, 206340);