#6389351

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,3,4,2)(6,9)(7,11);(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240753, #72257006, #72257130
maximal normal subgroup of	#72240753, #72257006, #72257130
maximal quotient of	#72240769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296547);