#6380906

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,26)(22,27)(24,28);(13,15,17,24)(14,18,19,27)(16,22,26,23)(20,28,25,21);(6,9)(10,12);(14,19)(15,22)(16,26)(18,21)(23,24)(27,28);(5,7,8,11)(6,10,9,12)(14,19)(18,27)(20,25)(21,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,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, 6288102);