#6817455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724651);