#6677582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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
prime radical	C₂×#3109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,26,30)(23,27,33,36)(24,28,25,29)(31,34,32,35);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,33)(22,27,30,36)(24,32,25,31)(28,35,29,34);(2,4)(23,31)(24,25)(27,34)(28,29)(32,33)(35,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,26,25)(22,28,30,29)(23,32,33,31)(27,35,36,34);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,31)(24,25)(27,34)(28,29)(32,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283144, #72283199, #72283292, …
maximal normal subgroup of	#72283144, #72283199, #72283292, …
maximal quotient of	#72311823, #72311905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584778);