#421072

order	512
nilpotency class	3
isoclinism	isoclinic class of #421072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15283
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328268);