#7186540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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)(8,16)(11,17)(15,20)(18,19)(21,22)(23,31)(24,29)(25,27)(26,38)(28,35)(30,32)(33,44)(34,43)(36,40)(37,49)(39,45)(41,48)(42,46)(47,51)(50,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,27)(24,36)(28,39)(30,38)(31,43)(33,47)(37,46)(48,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,32,45,25,34,26,35)(22,28,38,43,27,39,30,31)(24,33,46,52,36,47,37,48)(29,41,49,51,40,50,42,44);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,29)(23,33)(25,36)(26,37)(27,40)(28,41)(30,42)(31,44)(32,46)(34,47)(35,48)(38,49)(39,50)(43,51)(45,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(24,36)(29,40)(33,47)(37,46)(41,50)(42,49)(44,51)(48,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093736);