#6376367

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283563);