#4397207

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3018 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)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(10,11)(14,15)(18,22)(19,26)(24,28)(27,32);(1,2,4,3)(5,6)(7,8)(12,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(1,3,4,2)(9,11,13,10)(12,15,16,14)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(9,12)(10,14)(11,15)(13,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4304403);