#229535

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	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

684 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)(5,6,8,11,9,10,12,7)(13,14)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(1,3,4,2)(6,7)(8,12)(10,11)(15,23)(19,27)(22,28)(24,26);(6,10)(7,11)(13,15,18,23)(14,19,21,27)(16,24,17,26)(20,28,25,22);(5,7,8,6,9,11,12,10)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=ei,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	#72309471, #72309507, #72309597
maximal normal subgroup of	#72309471, #72309507, #72309597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136731);