#105292

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,d^a=dh,f^a=fh,c^b=ci

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12488);