#6380528

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240755, #72257004, #72257026
maximal normal subgroup of	#72240755, #72257004, #72257026
maximal quotient of	#72240765, #72308259

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287724);