#225254

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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)(3,5)(4,6)(7,8)(9,10,15,16,14,11,13,12)(18,22,24,28)(23,25,30,31);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,15,16,13)(17,18)(19,25)(20,28)(21,24)(22,29)(23,27)(26,31)(30,32);(2,6)(5,8)(10,11)(12,16)(19,26)(23,30)(25,31)(27,32);(9,12,13,11,14,16,15,10)(17,19,20,27,21,26,29,32)(18,23,22,25,24,30,28,31)

Polycyclic presentation:

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

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, 132450);