#212921

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309438, #72309510, #72309562, …
maximal normal subgroup of	#72309438, #72309510, #72309562, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120117);