#121836

order	512
nilpotency class	3
isoclinism	isoclinic class of #121835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,9)(4,7,12,14)(5,8)(6,11)(10,13,16,15)(17,18)(19,26)(20,23)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,21,29)(18,22,25,31)(20,27,30,28)(23,26,24,32);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(18,23)(19,27)(20,30)(22,31)(24,25)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,f^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, 29032);