#297434

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39358	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204630);