#357712

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264908);