#497116

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#127	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404312);