#297325

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	#36739	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1259 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)(5,6,9,11)(7,8,12,10)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(1,2,4,3)(6,11)(8,10)(13,14,17,19)(15,18,22,25)(16,20,23,26)(21,24,27,28);(5,7,10,6,9,12,8,11)(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(1,3,4,2)(5,8,9,10)(6,7,11,12)(14,19)(18,25)(20,26)(24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204521);