#354954

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216198
maximal normal subgroup of	#72216198
maximal quotient of	#72215773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262150);