#301958

order	512
nilpotency class	3
isoclinism	isoclinic class of #301949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62269	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

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, 209154);