#6463005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂²×#496	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2546 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)(7,12)(8,10)(9,11)(13,14,16,20,17,19,25,24)(15,21,23,28,22,18,26,27);(7,8)(10,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,27)(26,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,17,22)(14,18,19,21)(16,23,25,26)(20,27,24,28);(5,8,9,7)(6,10,11,12)(14,19)(18,21)(20,24)(27,28);(15,22)(18,21)(23,26)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308686, #72308698, #72308809
maximal normal subgroup of	#72308686, #72308698, #72308809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370201);