#1494867

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2872 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,26)(25,27);(5,7,9,8)(6,10,12,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,28);(5,7,9,8)(6,11,12,10)(13,16,19,17)(14,15,20,18)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,17,19,16)(14,15,20,18)(21,22)(23,24)(25,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,18,19,15)(14,17,20,16)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|b^2=fh,c^2=fgh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267678
maximal normal subgroup of	#72267678
maximal quotient of	#72267673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1402063);