#253300

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14)(3,9,18,34)(4,19,24,39)(5,11,26,17)(6,27,31,12)(7,13,32,42)(10,28,37,23)(15,43,45,60)(16,46,40,29)(20,51,35,58)(21,54,55,53)(22,50,57,64)(25,49,59,38)(30,48,62,41)(33,47,63,44)(36,61,52,56);(1,3,7,17)(2,9,13,5)(4,20)(6,16,30,49)(8,18,32,11)(10,33)(12,29,41,25)(14,34,42,26)(15,22)(19,43)(21,36)(23,53)(24,35)(27,46,48,59)(28,54)(31,40,62,38)(37,63)(39,60)(44,56)(45,57)(47,61)(50,51)(52,55)(58,64);(1,4,8,24)(2,10,14,37)(3,15,18,45)(5,21,26,55)(6,28,31,23)(7,22,32,57)(9,33,34,63)(11,35,17,20)(12,39,27,19)(13,36,42,52)(16,47,40,44)(25,58,59,51)(29,60,46,43)(30,61,62,56)(38,53,49,54)(41,64,48,50);(1,5,7,9)(2,11,13,18)(3,14,17,42)(4,21,22,33)(6,29,30,25)(8,26,32,34)(10,35,36,45)(12,40,41,38)(15,37,20,52)(16,48,49,27)(19,47,50,54)(23,43,56,51)(24,55,57,63)(28,60,61,58)(31,46,62,59)(39,44,64,53)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,g^b=gh,e^c=eh,f^c=fi,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, 160496);