#414742

order	512
nilpotency class	3
isoclinism	isoclinic class of #394253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1622 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)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,18)(17,24)(20,23)(21,27)(22,26)(25,28);(14,18)(15,21,22,25)(19,27,26,28)(20,24);(2,4)(13,15)(14,19)(16,22)(17,25)(18,26)(20,27)(21,23)(24,28);(5,7,8,12)(6,9,10,11)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224745, #72224758
maximal normal subgroup of	#72224745, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321938);