#7837473

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238144, #72238660, #72257158, …
maximal normal subgroup of	#72238144, #72238660, #72257158, …
maximal quotient of	#72263428

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7744669);