#6447643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3383 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305368, #72305370, #72305376
maximal normal subgroup of	#72305368, #72305370, #72305376
maximal quotient of	#72305646, #72305798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354839);