#7185556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(6,10)(8,11)(13,14)(15,20)(16,26)(17,22)(18,33)(19,23)(21,28)(24,37)(25,34)(27,43)(29,36)(30,44)(31,39)(32,40)(35,42)(38,41);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,25,32,31)(21,26,37,36)(22,34,40,39)(27,41,30,42)(35,43,38,44);(5,7)(6,11)(8,10)(9,12)(13,16,18,28,19,29,15,24)(14,21,20,26,23,37,33,36)(17,30,31,41,32,27,25,42)(22,38,34,44,40,35,39,43);(2,4)(6,10)(7,12)(13,17,19,32)(14,22,23,40)(15,25,18,31)(16,27,29,30)(20,34,33,39)(21,35,37,38)(24,41,28,42)(26,43,36,44);(5,8,9,11)(6,7,10,12)(13,15,19,18)(14,20,23,33)(16,28,29,24)(17,31,32,25)(21,36,37,26)(22,39,40,34)(27,41,30,42)(35,43,38,44)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092752);