#7197638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,24,39,23,28,22,29)(19,30,38,25,32,45,21,27)(20,26,40,48,35,44,36,43)(31,46,34,41,33,47,37,42);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19)(18,25)(20,33)(21,24)(22,38)(23,32)(26,42)(27,28)(29,45)(30,39)(31,35)(34,40)(36,37)(41,44)(43,46)(47,48);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,41)(27,42)(28,44)(29,43)(30,46)(32,33)(34,38)(39,48)(45,47);(3,8)(5,7)(10,11)(14,15)(17,21,24,19,23,38,22,32)(18,27,39,30,28,25,29,45)(20,34,40,33,35,37,36,31)(26,41,48,47,44,42,43,46);(9,12,13,16)(10,14,11,15)(18,28)(19,32)(20,35)(21,38)(29,39)(36,40)(41,42)(46,47)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=fgi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104834);