#331572

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
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×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238768);