#7604315

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂×#3341	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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)(5,6,9,8)(7,10,12,11)(13,14)(15,22)(16,21)(17,31)(18,33)(19,24)(20,26)(23,32)(25,36)(27,38)(28,35)(29,42)(30,43)(34,40)(37,41)(39,44);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,26,19,27,32,17)(14,20,23,22,24,31,33,38)(16,25,30,40,28,39,41,29)(21,34,37,36,35,42,43,44);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16,19,28)(14,21,24,35)(15,25,27,39)(17,29,26,40)(18,30,32,41)(20,34,31,42)(22,36,38,44)(23,37,33,43);(1,3,4,2)(7,12)(10,11)(16,28)(21,35)(25,39)(29,40)(30,41)(34,42)(36,44)(37,43);(13,17,32,27,19,26,18,15)(14,22,33,20,24,38,23,31)(16,29,41,39,28,40,30,25)(21,36,43,34,35,44,37,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=hi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511511);