#7968470

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

8425 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)(5,6,9,10)(7,11,12,8)(14,19)(15,21)(16,25)(24,28);(1,2)(3,4)(6,10)(7,12)(14,19)(17,26)(20,22)(24,28);(5,7)(6,11)(8,10)(9,12)(13,14,18,19)(15,22,21,20)(16,26,25,17)(23,24,27,28);(5,8)(6,12)(7,10)(9,11)(13,15,18,21)(14,20,19,22)(16,23,25,27)(17,24,26,28);(5,8)(6,7)(9,11)(10,12)(13,16,18,25)(14,17,19,26)(15,23,21,27)(20,24,22,28);(5,7)(6,11)(8,10)(9,12)(13,17)(14,16)(15,24)(18,26)(19,25)(20,23)(21,28)(22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279656, #72282769, #72284614, …
maximal normal subgroup of	#72279656, #72282769, #72284614, …
maximal quotient of	#72308755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875666);