#392426

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299622);