#246424

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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	C₂×C₄²	C₂⁴
prime radical	#59126	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264264, #72264312, #72264318
maximal normal subgroup of	#72264264, #72264312, #72264318

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153620);