#386837

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fg,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294033);