#6457673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35528	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2381 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226911, #72226919, #72226944
maximal normal subgroup of	#72226911, #72226919, #72226944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364869);