#7489100

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396296);