#404296

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10)(11,21,22,16)(12,24,25,17)(13,27)(14,30)(15,19)(18,29)(20,34)(23,37,31,38)(26,39,32,36)(28,35)(33,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,13)(14,31)(15,22)(17,35)(18,36)(19,21)(20,38)(23,34)(24,27)(25,28)(26,33)(29,32)(30,37)(39,40);(2,5)(6,8)(9,12)(10,17)(11,13)(14,32)(15,25)(16,35)(18,38)(19,24)(20,36)(21,27)(22,28)(23,33)(26,34)(29,31)(30,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,12)(14,33)(15,28)(16,36)(17,38)(19,40)(20,35)(21,39)(22,25)(23,32)(24,37)(26,31)(27,30)(29,34)

Polycyclic presentation:

2^9|a^2=ef,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311492);