#494898

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,18)(8,19)(9,15)(10,20)(11,26)(12,21)(13,17)(14,23)(16,27)(22,33)(24,35)(25,31)(28,30)(29,36)(32,34);(1,3,4,2)(5,7,18,6,12,20,11,15,13,16,27,17,9,26,10,21)(8,19,32,14,24,33,28,29,25,31,36,30,22,35,23,34);(5,8,10,23,9,22,27,36,13,25,11,28,12,24,18,32)(6,14,7,19,21,34,26,35,17,30,16,31,15,29,20,33);(5,9,13,12)(6,15,17,21)(7,20,16,26)(8,22,25,24)(10,27,11,18)(14,29,30,34)(19,33,31,35)(23,36,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250071, #72250079, #72250107, …
maximal normal subgroup of	#72250071, #72250079, #72250107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402094);