#617163

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330809, #72330847, #72330880, …
maximal normal subgroup of	#72330809, #72330847, #72330880, …
maximal quotient of	#72332544, #72333307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524359);