#7607122

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,26)(19,29,31,27)(20,32,34,25)(21,24,38,42)(22,39,40,28)(30,46,33,44)(35,41,37,43)(36,45,47,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,21,23,31,40,38)(18,24,28,29,26,42,39,27)(20,33,36,35,34,30,47,37)(25,43,45,46,32,41,48,44);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,34)(18,25,26,32)(19,30,31,33)(21,37,38,35)(22,36,40,47)(24,41,42,43)(27,46,29,44)(28,45,39,48);(2,6)(5,7)(11,16)(14,15)(18,26)(20,34)(24,42)(27,29)(28,39)(30,33)(35,37)(36,47);(2,6)(5,7)(17,21,22,31,23,38,40,19)(18,27,28,24,26,29,39,42)(20,35,36,30,34,37,47,33)(25,44,45,43,32,46,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7514318);