#392594

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3478 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)(5,6)(7,8)(9,10,14,17,15,12,13,18)(11,19,22,16,21,24,23,20);(2,4)(9,11,14,22,15,21,13,23)(10,16,17,24,12,20,18,19);(9,12)(10,15)(11,20)(13,17)(14,18)(16,21)(19,22)(23,24);(6,8)(11,21)(16,20)(19,24)(22,23)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,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	#72286571
maximal normal subgroup of	#72286571
maximal quotient of	#72286591, #72286714

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299790);