#7307248

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

1335 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)(25,26,31,37)(27,39,43,33)(28,45,29,35)(30,38,32,36)(34,47,52,46)(40,49,41,54)(42,50,44,51)(48,55,53,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,31,43)(26,33,37,39)(28,40,29,41)(30,42,32,44)(34,48,52,53)(35,49,45,54)(36,50,38,51)(46,55,47,56);(1,3)(2,5)(4,8)(6,7)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,46,32,47)(33,48,39,53)(35,36,45,38)(42,55,44,56)(49,50,54,51);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,31,28)(26,35,37,45)(27,41,43,40)(30,46,32,47)(33,49,39,54)(34,36,52,38)(42,55,44,56)(48,50,53,51);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,46,29,47)(33,50,39,51)(34,45,52,35)(40,55,41,56)(48,54,53,49)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7214444);