#7306994

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,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, 7214190);