#396432

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305687, #72305704, #72305705, …
maximal normal subgroup of	#72305687, #72305704, #72305705, …
maximal quotient of	#72322024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303628);