#6368672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1553 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306663, #72306696, #72306698, …
maximal normal subgroup of	#72306663, #72306696, #72306698, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275868);