#368070

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂²×C₄⋊C₄
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1735 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)(9,10,14,17,15,11,13,19)(12,20,23,18,24,21,22,16);(2,4)(9,11,14,19,15,10,13,17)(12,21,23,16,24,20,22,18);(6,8)(9,12,13,22,15,24,14,23)(10,16,17,20,11,18,19,21);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320920, #72320931, #72321009
maximal normal subgroup of	#72320920, #72320931, #72321009
maximal quotient of	#72321193, #72321474

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275266);