#6447741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318312, #72318405, #72318868, …
maximal normal subgroup of	#72318312, #72318405, #72318868, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354937);