#805032

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304048, #72304075, #72304113
maximal normal subgroup of	#72304048, #72304075, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 712228);