#6380660

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240755, #72256856, #72256942, …
maximal normal subgroup of	#72240755, #72256856, #72256942, …
maximal quotient of	#72240763, #72308242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287856);