#298820

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224774, #72224779, #72224781
maximal normal subgroup of	#72224774, #72224779, #72224781

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206016);