#299078

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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=eh,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, 206274);