#2831838

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2664 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326678, #72326716, #72326731
maximal normal subgroup of	#72326678, #72326716, #72326731
maximal quotient of	#72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2739034);