#414950

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1404 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=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, 322146);