#7514807

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	#36078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 7422003);