#299215

order	512
nilpotency class	3
isoclinism	isoclinic class of #299195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

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, 206411);