#389294

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

998 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=efhi,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,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, 296490);