#120846

order	512
nilpotency class	3
isoclinism	isoclinic class of #120844
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

373 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:

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

Polycyclic presentation:

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

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, 28042);