#6677740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284243, #72284625, #72307914, …
maximal normal subgroup of	#72284243, #72284625, #72307914, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584936);