#211371

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=i,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118567);