#7514710

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2717 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)(13,14,17,19)(15,23,26,37)(16,27,30,39)(18,32,31,22)(20,33,35,24)(21,34,36,28)(25,38,29,40)(41,42,44,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(16,28)(18,31)(20,37)(21,39)(23,35)(26,33)(27,36)(30,34)(38,40)(42,43);(13,15)(14,20)(16,29)(17,26)(18,33)(19,35)(21,40)(22,37)(23,32)(24,31)(25,30)(27,44)(28,43)(34,42)(36,38)(39,41);(2,4)(13,16,17,30)(14,21,19,36)(15,25,26,29)(18,34,31,28)(20,38,35,40)(22,39,32,27)(23,41,37,44)(24,42,33,43);(6,10)(9,11)(14,19)(15,26)(21,36)(22,32)(24,33)(25,29)(27,39)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^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,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421906);