#382536

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317251, #72317286, #72317290
maximal normal subgroup of	#72317251, #72317286, #72317290

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289732);