#6386308

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3164 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,26)(24,27)(25,29)(28,32)(30,33)(31,35)(34,36);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20);(6,14)(11,15)(12,13)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293504);