#6460029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2534 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)(5,6,8,9)(7,10,11,12)(13,14,18,21,19,22,17,15)(16,20,26,28,27,24,25,23);(6,9)(10,12)(13,15)(14,17)(16,24)(18,22)(19,21)(20,27)(23,25)(26,28);(1,2,3,4)(13,16,18,26,19,27,17,25)(14,20,21,28,22,24,15,23);(5,7,8,11)(6,10,9,12)(13,14,18,21,19,22,17,15)(16,20,26,28,27,24,25,23);(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,28,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307409, #72307418, #72307552
maximal normal subgroup of	#72307409, #72307418, #72307552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367225);