#397913

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2248 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310602, #72310611, #72310626
maximal normal subgroup of	#72310602, #72310611, #72310626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305109);