#347190

order	512
nilpotency class	3
isoclinism	isoclinic class of #346934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63011	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286053, #72286264, #72286397
maximal normal subgroup of	#72286053, #72286264, #72286397

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254386);