#254512

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	#29620	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

914 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,30,31)(26,28,29,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,28,30,32)(21,27)(22,25)(24,29,31,26);(4,11)(6,12)(7,14)(9,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,28)(25,32)(27,29);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=hi,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, 161708);