#1013935

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1948 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 921131);