#6388763

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,31)(24,35)(25,29)(26,39)(27,33)(28,37)(30,38)(32,47)(34,42)(36,44)(40,43)(41,51)(45,49)(46,50)(48,52);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23)(22,27)(24,32)(25,33)(26,34)(28,41)(29,31)(30,42)(35,52)(36,48)(37,49)(38,50)(39,46)(40,51)(43,45)(44,47);(21,24,25,36)(22,28,29,43)(23,32,33,48)(26,40,38,37)(27,41,31,45)(30,44,39,35)(34,51,50,49)(42,47,46,52);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(23,33)(27,31)(32,48)(34,50)(41,45)(42,46)(47,52)(49,51);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(22,29)(23,33)(28,43)(30,39)(32,48)(34,50)(35,44)(49,51)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295959);