#4610000

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332126, #72332147, #72332156
maximal normal subgroup of	#72332126, #72332147, #72332156
maximal quotient of	#72332880

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4517196);