#6684292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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×#3297	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256945, #72256965, #72257039
maximal normal subgroup of	#72256945, #72256965, #72257039
maximal quotient of	#72264430, #72264528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591488);