#324731

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fhi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231927);