#4388795

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2812	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3476 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,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, 4295991);