#224113

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×C₂²⋊C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dei,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131309);