#2063762

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1970958);