#245244

order	512
nilpotency class	3
isoclinism	isoclinic class of #245225
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#2403	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1390 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)(22,28)(25,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32);(6,12)(13,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20)(18,23)(19,25)(21,27)(22,29)(24,30)(26,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341043, #72341126, #72341143, …
maximal normal subgroup of	#72341043, #72341126, #72341143, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152440);