#347204

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	#63037	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2205 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,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	#72268187, #72268246
maximal normal subgroup of	#72268187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254400);