#229524

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309459, #72309491, #72309597
maximal normal subgroup of	#72309459, #72309491, #72309597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136720);