#192307

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1334 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99503);