#6380505

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2270 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)(8,12)(9,11)(13,14,18,21)(15,22,17,19)(16,20,24,27)(23,28,25,26);(5,7,9,8)(6,10,11,12)(14,19)(15,17)(16,23)(20,27)(21,22)(24,25);(5,8,9,7)(6,10,11,12)(13,15,18,17)(14,19,21,22)(16,23,24,25)(20,26,27,28);(13,16,18,24)(14,20,21,27)(15,23,17,25)(19,26,22,28);(14,21)(19,22)(20,27)(26,28)

Polycyclic presentation:

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

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, 6287701);