#7688154

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315080, #72315167, #72315191
maximal normal subgroup of	#72315080, #72315167, #72315191

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595350);