#412524

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

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, 319720);