#7148412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055608);