#386811

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294007);