#4063969

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

2562 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326805, #72326853, #72326884
maximal normal subgroup of	#72326805, #72326853, #72326884
maximal quotient of	#72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3971165);