#298834

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1069 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)(7,13)(8,17)(9,11)(10,15)(12,14)(16,19)(18,20)(21,22,25,27)(23,31,33,36)(24,30,26,29)(28,32,35,34);(1,2)(3,4)(5,7,8,15,9,16,14,20)(6,10,12,13,11,18,17,19)(22,27)(28,35)(29,30)(31,36);(2,4)(6,11)(10,18)(12,17)(13,19)(21,23,24,32,25,33,26,34)(22,28,29,36,27,35,30,31);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,36,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,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=dh

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, 206030);