#6448950

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2602 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)(7,10)(8,9)(11,12)(13,14,17,22,18,20,16,21)(15,23,24,27,25,28,26,19);(1,2)(3,4)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(2,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,22)(23,28);(13,16,18,17)(14,21,20,22)(15,24,25,26)(19,23,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280728, #72280745, #72310716, …
maximal normal subgroup of	#72280728, #72280745, #72310716, …
maximal quotient of	#72312973, #72313391

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356146);