#6400710

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,25)(16,20)(17,22)(18,24)(19,23)(21,27)(26,28);(5,7,10,6)(8,11,9,12)(13,15,18,16)(14,20,24,25)(17,27,19,26)(21,22,28,23);(13,16,18,15)(14,21,24,28)(17,27,19,26)(20,22,25,23);(5,8,10,9)(6,11,7,12)(13,17,18,19)(14,22,24,23)(15,26,16,27)(20,21,25,28);(1,3,4,2)(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,23,24,22)(15,26,16,27)(20,28,25,21)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307906);