#6374392

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281588);