#7542122

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449318);