#6368670

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

1536 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)(17,18,23,27)(19,30,33,45)(20,26,35,44)(21,29,22,28)(24,37,41,34)(25,38,42,36)(31,39,48,40)(32,46,47,43);(1,2)(3,8)(4,7)(5,14)(6,10)(9,11)(12,16)(13,15)(17,19,23,33)(18,24,27,41)(20,31,35,48)(21,37,22,34)(25,39,42,40)(26,43,44,46)(28,45,29,30)(32,38,47,36);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,14)(10,16)(17,20)(18,25)(19,31)(21,36)(22,38)(23,35)(24,39)(26,28)(27,42)(29,44)(30,46)(32,37)(33,48)(34,47)(40,41)(43,45);(1,3)(2,9)(4,11)(5,13)(6,12)(7,16)(8,14)(10,15)(17,20)(18,26)(19,32)(21,38)(22,36)(23,35)(24,40)(25,28)(27,44)(29,42)(30,46)(31,37)(33,47)(34,48)(39,41)(43,45);(1,4,6,5)(2,7,10,14)(3,11,12,13)(8,9,16,15)(18,27)(19,33)(20,35)(28,29)(34,37)(36,38)(39,40)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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	#72306688, #72306698, #72306708
maximal normal subgroup of	#72306688, #72306698, #72306708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275866);