#7307434

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1328 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)(25,26,31,36)(27,32,38,48)(28,40,29,34)(30,44,45,35)(33,43,50,41)(37,47,42,51)(39,54,53,49)(46,55,56,52);(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)(26,32)(28,37)(29,42)(30,39)(31,38)(33,46)(34,51)(35,49)(36,48)(40,47)(41,52)(43,55)(44,54)(45,53)(50,56);(1,3)(2,5)(4,8)(6,7)(25,28,31,29)(26,33,36,50)(27,37,38,42)(30,43,45,41)(32,46,48,56)(34,44,40,35)(39,55,53,52)(47,49,51,54);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,31,28)(26,34,36,40)(27,37,38,42)(30,43,45,41)(32,47,48,51)(33,44,50,35)(39,52,53,55)(46,49,56,54);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(27,38)(28,29)(32,48)(33,50)(34,40)(39,53)(41,43)(49,54)

Polycyclic presentation:

2^9|a^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=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7214630);