#7785333

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

7685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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	#72326930
maximal normal subgroup of	#72326930
maximal quotient of	#72326998, #72329595, #72336620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7692529);