#224936

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=ei,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132132);