#6379686

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,f^b=fi,f^d=fi

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, 6286882);