#110498

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 17694);