#7181108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 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)(21,22,28,33)(23,29,38,46)(24,31,25,40)(26,32,27,42)(30,41,47,39)(34,44,35,49)(36,45,37,51)(43,50,52,48);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,28,38)(22,29,33,46)(24,34,25,35)(26,36,27,37)(30,43,47,52)(31,44,40,49)(32,45,42,51)(39,48,41,50);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,28,25)(22,30,33,47)(23,34,38,35)(26,39,27,41)(29,43,46,52)(31,32,40,42)(36,48,37,50)(44,45,49,51);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,28,24)(22,31,33,40)(23,35,38,34)(26,39,27,41)(29,44,46,49)(30,32,47,42)(36,48,37,50)(43,45,52,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,33,42)(23,36,38,37)(24,39,25,41)(29,45,46,51)(30,40,47,31)(34,48,35,50)(43,49,52,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088304);