#375309

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1974 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)(9,10,13,18,14,19,23,17)(11,16,21,15,22,24,12,20);(5,6,7,8)(9,11,13,21,14,22,23,12)(10,15,17,16,19,20,18,24);(1,2,3,4)(6,8)(9,10)(11,20)(12,16)(13,17)(14,19)(15,22)(18,23)(21,24);(9,12,23,22,14,21,13,11)(10,16,18,15,19,24,17,20)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=fhi,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226660, #72226670
maximal normal subgroup of	#72226660, #72226670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282505);