#2806508

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327692, #72327796, #72327870
maximal normal subgroup of	#72327692, #72327796, #72327870
maximal quotient of	#72330141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2713704);