#7670671

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

7788 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305258, #72305266, #72305286
maximal normal subgroup of	#72305258, #72305266, #72305286
maximal quotient of	#72306208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577867);