#562693

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329554, #72329562, #72329586
maximal normal subgroup of	#72329554, #72329562, #72329586
maximal quotient of	#72331431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 469889);