#121838

order	512
nilpotency class	3
isoclinism	isoclinic class of #121835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh,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, 29034);