#3331575

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15947	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3214 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=ghi,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329171, #72329189, #72329233
maximal normal subgroup of	#72329171, #72329189, #72329233
maximal quotient of	#72331381

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3238771);