#3348639

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3049 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278942
maximal normal subgroup of	#72278942
maximal quotient of	#72279004, #72299397, #72299666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3255835);