#359058

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1908 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)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(2,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,23)(22,24);(6,8)(9,10)(11,14)(12,15)(13,17)(16,18)(19,21)(20,23)(22,24);(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320920, #72320984
maximal normal subgroup of	#72320920, #72320984
maximal quotient of	#72321162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266254);