#2832567

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=fg,e^2=gh,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	#72266508, #72266520, #72266523, …
maximal normal subgroup of	#72266508, #72266520, #72266523, …
maximal quotient of	#72334830

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2739763);