#414612

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	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1992 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)(3,4)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,27,28,31)(20,26,21,24)(23,30,32,29);(2,4)(5,6)(7,8)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,32)(26,27);(6,8)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(9,11,12,16)(10,13,14,15)(18,25)(19,28)(24,26)(29,30)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,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.

Relationships

maximal subgroup of	#72265033, #72265061, #72265102
maximal normal subgroup of	#72265033, #72265061, #72265102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321808);