#295022

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fh,d^2=ei,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, 202218);