#4423123

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4330319);