#358753

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265949);