#186068

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#3535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2574 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)(7,8,11,12)(9,10)(13,14,19,21,18,22,17,28)(15,23,25,16,20,26,24,27);(2,4)(5,7)(6,8)(9,11)(10,12)(13,15,18,20)(14,16,22,27)(17,24,19,25)(21,26,28,23);(5,8)(6,7)(9,12)(10,11)(13,16)(14,20)(15,22)(17,26)(18,27)(19,23)(21,25)(24,28);(6,10)(8,12)(13,17,18,19)(14,21,22,28)(15,24,20,25)(16,26,27,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93264);