#498373

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=ehi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72225570, #72225573
maximal normal subgroup of	#72216402, #72225570, #72225573
maximal quotient of	#72226899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405569);