#370235

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1110 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277431);