#647825

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	#15123	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239704, #72239711, #72239781, …
maximal normal subgroup of	#72239704, #72239711, #72239781, …
maximal quotient of	#72250492, #72298476, #72320982

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 555021);