#380203

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 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)(5,6,8,7)(10,11)(14,15)(17,18)(19,23)(20,22)(21,24)(25,32)(26,29)(27,30)(28,31);(5,7,8,6)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(6,7)(9,11,13,10)(12,14,16,15)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(9,12,13,16)(10,14,11,15)(18,22)(19,26)(24,30)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,g^c=gi

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, 287399);