#2805714

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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

4333 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,20,23,24);(6,8)(9,10,12,14)(11,13,16,15)(18,22)(20,24);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,20,23,24);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24);(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,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	#72327324, #72327361, #72327383, …
maximal normal subgroup of	#72327324, #72327361, #72327383, …
maximal quotient of	#72330548, #72332764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2712910);