#299017

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	C₂×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eh,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

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, 206213);