#612619

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

13270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237100, #72237113, #72237115, …
maximal normal subgroup of	#72237100, #72237113, #72237115, …
maximal quotient of	#72264260, #72287873, #72287877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519815);