#234832

order	512
nilpotency class	3
isoclinism	isoclinic class of #234825
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142028);