#370069

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi

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, 277265);