#6357955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,e^b=ehi,d^c=di,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306780, #72306781, #72306783, …
maximal normal subgroup of	#72306780, #72306781, #72306783, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265151);