#7670675

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	#36123	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7288 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)(6,10)(11,12)(13,14)(15,23)(16,20)(17,27)(18,22)(19,25)(21,28)(24,26);(5,6)(7,11)(8,12)(9,10)(13,15,18,25)(14,19,22,23)(16,17,24,28)(20,21,26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,14,18,22)(15,23,25,19)(16,26,24,20)(17,21,28,27);(1,2)(3,4)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,26)(27,28);(5,8,9,7)(6,12,10,11)(13,16,18,24)(14,20,22,26)(15,17,25,28)(19,21,23,27);(13,17)(14,21)(15,24)(16,25)(18,28)(19,26)(20,23)(22,27)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^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	#72305240, #72305251, #72305286
maximal normal subgroup of	#72305240, #72305251, #72305286
maximal quotient of	#72305800, #72305812

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577871);