#7516240

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2011 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7423436);