#385998

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂×#2397	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2154 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)(7,8)(13,14,18,22,17,19,16,21)(15,20,23,27,24,26,25,28);(1,2)(3,4)(10,12)(14,19)(20,26)(21,22)(27,28);(6,8)(9,10,11,12)(13,15)(14,20)(16,25)(17,24)(18,23)(19,26)(21,28)(22,27);(1,3,4,2)(6,8)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,27,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293194);