#192431

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99627);