#303796

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,11)(12,15,16,13)(17,18,22,25)(19,23,27,31)(20,28,30,32)(21,26,29,24);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,27)(18,23,25,31)(20,26,30,24)(21,28,29,32);(1,3,5,4)(2,7,8,6)(9,12,13,10,14,16,15,11)(17,20,29,23,22,30,21,31)(18,19,26,32,25,27,24,28);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,16,11,12)(19,27)(20,30)(23,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efh,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=dhi

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, 210992);