#239806

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147002);