#246316

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37770	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1269 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153512);