#298132

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205328);