#248893

order	512
nilpotency class	3
isoclinism	isoclinic class of #248839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2847	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156089);