#147451

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216642, #72242475, C₄×#13138
maximal normal subgroup of	#72216642, #72242475, C₄×#13138
maximal quotient of	#72216573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54647);