#254505

order	512
nilpotency class	3
isoclinism	isoclinic class of #254491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

894 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,23)(19,26,21,24)(20,25,29,31)(27,28,30,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,22,21)(18,24,23,26)(20,28)(25,30)(27,31)(29,32);(4,11)(6,12)(7,14)(9,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,28)(26,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,22,19)(18,26,23,24)(20,27,29,30)(25,28,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161701);