#375186

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1694 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)(7,11)(8,10)(9,12)(13,14,18,21)(15,19,23,22)(16,17,20,26)(24,28,27,25);(5,7,8,12)(6,9,10,11)(13,15,17,24,18,23,26,27)(14,19,20,28,21,22,16,25);(2,4)(13,16,17,21,18,20,26,14)(15,22,24,28,23,19,27,25);(6,10)(9,11)(15,23)(19,22)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fhi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226660, #72226674
maximal normal subgroup of	#72226660, #72226674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282382);