#7670681

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

7433 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)(5,6,9,12)(7,10,8,11)(13,14,19,16)(15,23,24,20)(17,21,28,26)(18,27,25,22);(5,6)(7,10)(8,11)(9,12)(13,15,19,24)(14,20,16,23)(17,25,28,18)(21,27,26,22);(2,4)(15,24)(17,28)(20,23)(21,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,16,19,14)(15,23,24,20)(17,26,28,21)(18,27,25,22);(13,17)(14,21)(15,25)(16,26)(18,24)(19,28)(20,27)(22,23);(5,8,9,7)(6,11,12,10)(13,18,19,25)(14,22,16,27)(15,17,24,28)(20,21,23,26)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,f^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	#72305281, #72305286
maximal normal subgroup of	#72305281, #72305286
maximal quotient of	#72306012, #72306051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577877);