#238439

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1166 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)(5,6,10,12)(7,8,11,9)(13,14,18,22)(15,24,25,16)(17,21,19,23)(20,28,26,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,25)(19,26)(21,27)(22,24)(23,28);(1,3,4,2)(6,11)(7,12)(8,9)(13,16,18,24)(14,20,22,26)(15,21,25,23)(17,27,19,28);(13,17,18,19)(14,21,22,23)(15,20,25,26)(16,27,24,28)

Polycyclic presentation:

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

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, 145635);