#299002

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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

984 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)(7,13,16,20)(8,17,14,12)(10,18,19,15)(21,22)(23,28)(24,34)(25,35)(26,30)(27,32)(29,31)(33,36);(1,2)(3,4)(5,7,14,18,9,16,8,15)(6,10,17,20,11,19,12,13)(21,23,31,24,26,33,25,32)(22,27,35,36,30,34,29,28);(2,4)(6,11)(10,19)(12,17)(13,20)(21,23,25,32,26,33,31,24)(22,27,29,28,30,34,35,36);(5,7,14,18,9,16,8,15)(6,10,17,20,11,19,12,13)(21,24,31,33,26,32,25,23)(22,28,35,27,30,36,29,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fhi,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206198);