#7194152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,40)(36,47)(37,45)(39,51)(41,52)(42,48)(43,49)(46,53)(50,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,43,42)(33,38,47,52)(34,45,49,48)(39,50,54,53)(46,51,56,55);(1,2,5,4)(3,6,8,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,47,44,52)(29,39,42,53,43,54,37,50)(34,46,45,51,49,56,48,55);(1,4,5,2)(3,6,8,7)(12,19)(15,17)(20,24)(22,23)(25,29,31,43)(26,34,35,49)(27,37,30,42)(28,39,41,54)(32,45,44,48)(33,46,47,56)(36,50,40,53)(38,51,52,55);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,42,43,37)(33,38,47,52)(34,48,49,45)(39,53,54,50)(46,55,56,51)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fg,d^2=fi,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=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, 7101348);