#390395

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

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, 297591);