#4308738

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 25165824
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	#15172	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3086 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215934);