#394270

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2971 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)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24);(9,11,14,17)(10,15,18,13)(12,20)(16,23)(19,22)(21,24);(2,4)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(6,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,20,24,19)

Polycyclic presentation:

2^9|b^2=ei,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, #72310605, #72310626
maximal normal subgroup of	#72310602, #72310605, #72310626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301466);