#261196

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35166	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1386 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=hi,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168392);