#616978

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3956 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)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(6,8)(9,11,13,16)(10,14,15,12)(18,22)(19,25)(24,27)(26,31);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)

Polycyclic presentation:

2^9|b^2=g,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	#72330797, #72330805, #72330935
maximal normal subgroup of	#72330797, #72330805, #72330935
maximal quotient of	#72333159

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524174);