#196591

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1394 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)(5,6,9,12,10,11,8,7)(13,14,17,21,18,22,16,26)(15,20,25,27,23,28,24,19);(5,7,9,11,10,12,8,6)(13,15)(14,19,22,27)(16,24)(17,25)(18,23)(20,26,28,21);(2,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(5,8,10,9)(6,7,11,12)(15,23)(19,27)(20,28)(24,25)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,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, 103787);