#7173642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=i,e^2=hi,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080838);