#294402

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fh,d^2=e,f^2=h,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^c=gi

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, 201598);