#364270

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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)(14,19)(15,22)(16,17)(20,23)(21,26)(24,25);(5,6,8,9)(7,10,11,12)(13,14,18,21)(15,23,25,20)(16,26,17,19)(22,28,24,27);(6,9)(10,12)(13,15,17,22,18,25,16,24)(14,20,19,27,21,23,26,28);(5,7,8,11)(6,10,9,12)(13,14,18,21)(15,20,25,23)(16,26,17,19)(22,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271466);