#3859916

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3020 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333470, #72333479, #72333487
maximal normal subgroup of	#72333470, #72333479, #72333487

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3767112);