#400262

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14798
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 307458);