#196465

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103661);