#185254

order	512
nilpotency class	3
isoclinism	isoclinic class of #185249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

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, 92450);