#120975

order	512
nilpotency class	3
isoclinism	isoclinic class of #120974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fh,e^c=ehi,g^c=gh

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, 28171);