#226909

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=di

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, 134105);