#493201

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

633 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)(3,6)(4,5)(7,8)(9,10,14,21,15,17,24,22,16,11,18,23,13,20,12,19);(2,5)(6,8)(10,17,11,20)(12,24)(14,18)(19,23,22,21);(9,11,14,23,15,20,24,19,16,10,18,21,13,17,12,22);(1,3,4,7)(2,6,5,8)(9,12,13,18,16,24,15,14)(10,19,20,23,11,22,17,21)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227016, #72310153, #72310158, …
maximal normal subgroup of	#72227016, #72310153, #72310158, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400397);