#385613

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,19,22,24,30,26)(20,23,28,25,29,32,31,27);(1,2)(3,4)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,27)(22,26)(23,31)(24,30)(25,29)(28,32);(2,4)(9,11,13,12)(10,15,16,14)(17,20,22,29)(18,23,24,32)(19,25,26,27)(21,28,30,31);(9,12,13,11)(10,14,16,15)(18,19)(21,30)(23,25)(24,26)(27,32)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi

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, 292809);