#315192

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=efh,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222388);